On the nonexistence of Green's function and failure of the strong maximum principle

Given any Borel function $V : \Omega \to [0, +\infty]$ on a smooth bounded domain $\Omega \subset \mathbb{R}^{N}$, we establish that the strong maximum principle for the Schr\"odinger operator $-\Delta + V$ in $\Omega$ holds in each Sobolev-connected component of $\Omega \setminus Z$, where $Z \subset \Omega$ is the set of points which cannot carry a Green's function for $- \Delta + V$. More generally, we show that the equation $- \Delta u + V u = \mu$ has a distributional solution in $W_{0}^{1, 1}(\Omega)$ for a nonnegative finite Borel measure $\mu$ if and only if $\mu(Z) = 0$.